Real higher-order Weyl photonic crystal

Higher-order Weyl semimetals are a family of recently predicted topological phases simultaneously showcasing unconventional properties derived from Weyl points, such as chiral anomaly, and multidimensional topological phenomena originating from higher-order topology. The higher-order Weyl semimetal phases, with their higher-order topology arising from quantized dipole or quadrupole bulk polarizations, have been demonstrated in phononics and circuits. Here, we experimentally discover a class of higher-order Weyl semimetal phase in a three-dimensional photonic crystal (PhC), exhibiting the concurrence of the surface and hinge Fermi arcs from the nonzero Chern number and the nontrivial generalized real Chern number, respectively, coined a real higher-order Weyl PhC. Notably, the projected two-dimensional subsystem with kz = 0 is a real Chern insulator, belonging to the Stiefel-Whitney class with real Bloch wavefunctions, which is distinguished fundamentally from the Chern class with complex Bloch wavefunctions. Our work offers an ideal photonic platform for exploring potential applications and material properties associated with the higher-order Weyl points and the Stiefel-Whitney class of topological phases.


Note 1. Tight-binding model
As investigated in Ref. 1 , a symmetry-enforced spin-1 Weyl point (charge-2 triple point) located at the Γ point, in conjunction with a charge-2 Dirac point at R point, can be realized in a photonic crystal with space group (SG) No.198.The generators of the SG No. 198 encompass C2x, C3,111 and T operations.Intriguingly, we find that the C3,111 symmetry plays a pivotal role in the formation of the spin-1 Weyl point 2 , however, it is not necessary for the presence of the charge-2 Dirac point.
Hence, upon breaking the C3,111 symmetry of the photonic crystal, the charge-2 Dirac point remains intact, while the spin-1 Weyl point will be split into two real higher-order Weyl points.
The essential physics discussed here can be captured by a four-band tight-binding (TB) model, for which the structure is shown in Fig. S1.Each unit cell contains four atoms located at (0.45,0.2,0.2),(0.95,0. The perturbed band structure is shown in Fig. S2b, where the spin-1 Weyl point at the Γ point indeed transforms into two C-1 Weyl points at z-axis.Simultaneously, the kz = 0 plane become an insulator phase.We then show under which conditions the kz = 0 plane is a real Chern insulator.S1.The C2z eigenvalues are calculated from the tight-binding (TB) models.The C2z eigenvalues of a state is calculated as follow.At a C2z invariant point K, C2z eigenvalues of n th band is defined as is the α th orbit in Ri cell.In this basis, wave function can be expanded as cnα is the α th element of n th eigenvector of Hamiltonian H(k).By combining Eq. ( 1) and (3), one can get the expression of C2z eigenvalue of n th band as where C2zβα is the C2z matrix under the basis of     .This can be rewritten into a compact matrix No. 198 splits into two Weyl points and the kz = 0 plane is gapped (see Fig. S2b), the lowest two bands of the TB model in Fig. S2b at the Γ point must be Γ1 and Γ2, ensuring that the kz = 0 plane possesses a vanishing Chern number 3,4 .This is because the Chern number (C) of kz = 0 plane can be inferred by the C2z-eigenvalue at Γ, X, Y and M points 3 ( 1) ( ) ( ) ( ) ( ) where i  is C2z-eigenvalue at Γ, X, Y and M points on the i-th band.Meanwhile, we find that for the TB model depicted in Fig. S2b the lowest two bands at the S point are S3⊕ S4 with the C2zeigenvalue of (-1,-1).Then, according to the definition of real Chern number, one has vR = 1 for the kz = 0 plane.It should be noticed that for the TB model in Fig. S2b, the lowest two bands at S point also can be S1⊕ S2 under suitable model parameter, and in such case the real Chern number of the kz = 0 plane is zero.Since the kz = 0 plane is a real Chern insulator, it will have corner states, which also can be understood from topological quantum chemistry (TQC) [5][6][7][8] .The kz = 0 plane is a 2D system belonging to layer group No. 21.Analysis with TQC shows that, the 2D system is equivalent to that of two orbitals (Wannier centers) at the 2a Wyckoff position: (0,0,z) and (1/2,1/2,-z).Notice that the 2a site is not occupied by any atom of the TB model, as shown in Fig. S3.Generally, for a nanotube cutting through a Wannier center, it will have hinge states.However, it should be noticed that for certain that a nanotube with a triangular cross section (see Fig. S3) is most suitable, which is not only convenient for experiment realization but also exhibits a clean hinge state.

Note 2. Nontrivial Zak phase
Due to the C2z symmetry, the Zak phase along a line transverse the BZ is quantized, for which the value can also be inferred from the C2z-eigenvalue at two C2z-invariant points 9, 10 , as the C2z symmetry is identical to a mirror symmetry for a one-dimensional system.
Hence, the Zak phase Zx and Zy of all the |kz| < kwp plane of SG No. 19 can be obtained as According to C2z-eigenvalue, one has Zx = Zy = 1 for all the |kz| < kwp planes, which indicates the surface floating states between the two real higher-order Weyl points.

Note 3. Simulated results of the evolution from spin-1 Weyl point to real higher-order Weyl points.
First, we construct a photonic crystal featuring three C2 screw symmetry along the x, y, z axis, and a C3 rotational symmetry along the <111> axis.The 3D unit cell is shown in Fig. S4a.The simulated band structure is shown in Fig. S4b.As observed, the spin-1 Weyl point exists at Γ, Fig. S4c, and the photonic band structure is displayed in Fig. S4d.Evidently, the spin-1 Weyl point splits into two real higher-order Weyl points.However, the crystal structure shown in Fig. S4c is excessively intricate for fabrication purposes.Therefore, we optimize the structure while maintaining its symmetry unaltered.for measuring the bulk states, surface states, and hinge states respectively.

Note 6. Topological surface states on the (010) surface
The experimental configuration is depicted in Fig. S7a.The source is placed in the center of the (010) surface, and the probe is inserted into the sample to measure the field distributions on the surface.After applying a 2D transform to the measured field distribution, we obtain the surface dispersion, as illustrated in Fig. S7b.The measured surface dispersion along the high-symmetry line Z M X  − − − −  is displayed in Fig. S7b, and the Zak phase protected topological floating surface states is observed along Z − and X −.The grey (green) dots represent the bulk (surface) states respectively.The color map measures the energy density.Fig. S7c-f show the Fermi arc surface states on the (010) surface of the photonic crystal.In order to calculate the topological charges for the real Weyl point and the charge-2 Dirac point, we first track the evolution of Wannier centers on a sphere covering four kinds of band degenerate points.The sphere is discretized into a sequence of horizontal loops, from the north pole to the south pole of the sphere (i.e., the polar angle θ in spherical coordinates varies from 0 to π).
Then the Berry phase along these horizontal loops can be numerically calculated by employing the Wilson loop method, in which the wavefunctions are extracted from the COMSOL Multiphysics calculations.The Wannier centers (ϕ) are simply the trace of the Berry phase.Fig. S8 shows the Wannier centers for the lower band of the real higher-order Weyl point and the charge-2 Dirac point.For the real higher-order Weyl point and charge-2 Dirac point, the Wannier centers shift by 2 and 4 − respectively, revealing the topological charge is 1 and -2, respectively.For the surface dispersion, we have changed the supercell size from 4 to 15 unit cells, and the simulated results are displayed in Fig. S12a-d.One can see that the surface dispersion is convergent when the size reaches 8 unit cells.For the hinge dispersion, we change the supercell size from 3×3 to 11×11 unit cells, and the simulated results are displayed in Fig. S13a-d.One can see that the hinge dispersion is convergent when the size reaches 8×8 unit cells.

Fig. S2 a ,
Fig. S2 a, Band structure of TB model (1) in space group No.198.b, Perturbed band structure.The inserts show the corresponding 3D Brillouin zone.

where 2 ˆz
C is the C2z operator and nK  is the wave function of n th band at .For the TB model, the Hamiltonian and wave functions are written in basis   exists only one band representation for X (Y) point: X1 (Y1), which is doubly degenerate and possesses the C2z-eigenvalue of (1,-1).Besides, there are two band representations for M (S) point, with C2z-eigenvalues of (1,1) and (-1,-1).Hence, when the spin-1 Weyl point in SG

Fig. S3
Fig. S3 Top view of the crystal structure of the perturbed TB model with the unit cell being denoted by the gray solid lines.The brown balls denote the atoms of the TB model, and the red balls are the Wannier centers obtained from the kz = 0 plane.The dashed lines show the boundary of a nanotube.

Fig. S4 a , 6 Note 4 .
Fig. S4 a, Unit cell of 3D photonic crystal hosting a spin-1 Weyl point.b, Band structure of the photonic crystal shown in Fig. S4a.c, Unit cell of a 3D photonic crystal hosting two real higherorder Weyl points.d, Band structure of the photonic crystal shown in Fig. S4c.

Fig. S7 a ,
Fig. S7 a, Experimental set-up.b, Measured surface dispersion along the high-symmetry line Z M X  − − − −  .The grey (green) dots represent the simulated bulk (surface) states respectively.The color map measures the energy intensity.c-f, Measured surface iso-frequency contours from 11.4 GHz to 11.7 GHz.The grey (green) curves represent the bulk (surface) dispersions.

Fig. S8 1 R
Fig. S8 Evolution of the Wannier centers on the spheres enclosing real Weyl point and charge-2 Dirac point, using Wilson loop method.a, Charge-1 real higher-order Weyl point.b, Charge-2 Dirac point.

Fig. S9 a , 1 R
Fig. S9 a, Wilson loop spectrum calculated along <110> direction on the kz = 0 plane.There exists one crossing point at  = , indicating real Chern number

Fig. S11 a, b ,
Fig. S11 a,b, Fabricated samples made of AlSi10Mg and Cu, respectively.The red star represents the source location.c,d, Measured dispersions of AlSi10Mg sample and Cu sample, respectively.The green dots display the simulated dispersion of a PEC structure, and the blue line shows the light line.e, Simulated and measured field distributions at different frequencies.The left, middle, and right columns represent the results of the structures made of PEC, AlSi10Mg, and Cu, respectively

Fig. S12
Fig. S12 The simulated surface dispersion when the supercell size changes from 4 to 15 unit cells.a, The simulated surface dispersion when the supercell consists of 4 unit cells.b, The simulated surface dispersion when the supercell consists of 8 unit cells.c, The simulated surface dispersion when the supercell consists of 12 unit cells.d, The simulated surface dispersion when the supercell consists of 15 unit cells.The red (grey) curves represent the surface (bulk) dispersions.

Fig. S13
Fig. S13 The simulated hinge dispersion when the supercell size changes from 3×3 to 11×11 unit cells.a, The simulated hinge dispersion when the supercell consists of 3×3 unit cells.b, The simulated surface dispersion when the supercell consists of 5×5 unit cells.c, The simulated surface dispersion when the supercell consists of 8×8 unit cells.d, The simulated surface dispersion when the supercell consists of 11×11 unit cells.The red (grey) curves represent the hinge (bulk) dispersions.

Table . S1
C2z-eigenvalue of the band representations at Γ, X, Y and M points.The Γ4 of SG No. 198 is transformed into three different band representations: Γ2, Γ3 and Γ4 of SG No. 19 upon symmetry breaking.The M point in SG No. 198 is labeled as S point in SG No. 19.